Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

On the Heisenberg-Lie Algebra and Some Non-Hermitian Operators in Oscillatorlike Developments

Jules Beckers, N. Debergh

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

After a few generalities we compare fundamental quantum mechanics applied to the harmonic oscillator with unusual oscillatorlike developments dealing with non-hermitian operators, this latter aspect exploiting in particular the property of subnormality. In this last context we can also restore the hermiticity of the Hamiltonian operator and discover a nice property of the new scalar product. General constructions of oscillatorlike Hamiltonians are also considered and new ideas connected with Heisenberg relations are presented.

Related papers

NON-HERMITIAN OSCILLATOR-LIKE HAMILTONIANS AND λ-COHERENT STATES REVISITED

Jules Beckers, N. Debergh, José F. Cariñena +1 more

Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.

arXiv:hep-th/0102004•Modern Physics Letters A•Jan 20, 2001•14 citations

A non-Hermitian oscillator Hamiltonian and su(1,1): a way towards generalizations

C Quesne

The family of metric operators, constructed by Musumbu et al ( 2007 J. Phys. A: Math. Theor. 40 F75 ), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily -symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.

arXiv:0705.2868•Journal of Physics A Mathematical and Theoretical•Jul 12, 2007•34 citations

A non-Hermitian oscillator Hamiltonian and su(1, 1): a way towards generalizations

C. Quesne

The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)

•Nov 21, 2012•12 citations

Pseudo-Hermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscillators

Alexandre M. Gavrilik, І. І. Качурик

The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.

arXiv:1808.04714•Modern Physics Letters A•Dec 14, 2018•1 citation

COHERENT STATES AND SCHWINGER MODELS FOR PSEUDO GENERALIZATION OF THE HEISENBERG ALGEBRA

Hossein Fakhri, B. Mojaveri, Alireza Dehghani

We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with [Formula: see text] and [Formula: see text] symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs of creation and annihilation operators which are [Formula: see text]-pseudo-Hermiticity and [Formula: see text]-anti-pseudo-Hermiticity of each other. The non-unitary Heisenberg algebra is represented by each of the pair of the operators in two different ways. Consequently, the coherent and the squeezed coherent states are calculated in two different approaches. Moreover, it is shown that the approach of Schwinger to construct the su(2), su(1, 1) and sp(4, ℝ) unitary algebras is promoted so that unitary algebras with more linearly dependent number of generators are made.

Modern Physics Letters A•Aug 20, 2009•4 citations

From nonrelativistic to relativistic properties of quantum harmonic oscillators

J. Beckers, Jean Ndimubandi

We investigate some possible connections between two recent approaches of so-called relativistic harmonic oscillators compatible with well known nonrelativistic properties. These approaches are not equivalent but can be related through simple arguments based on quantum deformations of oscillatorlike algebras when one-dimensional spatial developments are visited.

Physica Scripta•Jul 1, 1996•1 citation
After a few generalities we compare fundamental quantum mechanics applied to the harmonic oscillator with unusual oscillatorlike developments dealing with non-hermitian operators, this latter aspect exploiting in particular the property of subnormality. In this last context we can also restore the hermiticity of the Hamiltonian operator and discover a nice property of the new scalar product. General constructions of oscillatorlike Hamiltonians are also considered and new ideas connected with Heisenberg relations are presented.
The family of metric operators, constructed by Musumbu et al ( 2007 J. Phys. A: Math. Theor. 40 F75 ), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily -symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)
The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.
We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with [Formula: see text] and [Formula: see text] symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs of creation and annihilation operators which are [Formula: see text]-pseudo-Hermiticity and [Formula: see text]-anti-pseudo-Hermiticity of each other. The non-unitary Heisenberg algebra is represented by each of the pair of the operators in two different ways. Consequently, the coherent and the squeezed coherent states are calculated in two different approaches. Moreover, it is shown that the approach of Schwinger to construct the su(2), su(1, 1) and sp(4, ℝ) unitary algebras is promoted so that unitary algebras with more linearly dependent number of generators are made.